Mathematical Methods and its Applications

This course is a basic course offered to UG student of Engineering/Science background. It contains ODE,PDE, Laplace transforms, Z-transforms, Fourier series and Fourier transforms. It plays an important role for solving various engineering sciences problems.Therefore, it has tremendous applications in diverse fields in engineering sciences.INTENDED AUDIENCE : UG students of technical universities/colleges.PRE-REQUISITES : NilINDUSTRY SUPPORT : Nil

Week 1: Introduction to linear differential equations ,Linear dependence, independence and Wronskian of functions,Solution of second-order homogeneous linear differential equations with constant coefficients-I,Solution of second-order homogeneous linear differential equations with constant coefficients-II,Method of undetermined coefficientsWeek 2: Methods for finding Particular Integral for second-order linear differential equations with constant coefficients-I,Methods for finding Particular Integral for second-order linear differential equations with constant coefficients-II, Methods for finding Particular Integral for second-order linear differential equations with constant coefficients-III,Euler-Cauchy equations,Method of reduction for second-order, linear differential equationsWeek 3: Method of variation of parameters , Solution of second order differential equations by changing dependent variable, Solution of second order differential equations by changing independent variable,Solution of higher-order homogenous linear differential equations with constant coefficients, Methods for finding Particular Integral for higher-order linear differential equationsWeek 4: Formulation of Partial differential equations, Solution of Lagrange’s equation-I, Solution of Lagrange’s equation-II,Solution of first order nonlinear equations-I,Solution of first order nonlinear equations--IIWeek 5: Solution of first order nonlinear equations-III,Solution of first order nonlinear equations-IV,Introduction to Laplace transforms,Laplace transforms of some standard functions,Existence theorem for Laplace transformsWeek 6: Properties of Laplace transforms--I, Properties of Laplace transforms--II, Properties of Laplace transforms--III, Properties of Laplace transforms--IV, Convolution theorem for Laplace transforms--IWeek 7: Convolution theorem for Laplace transforms--II,Initial and final value theorems for Laplace transforms, Laplace transforms of periodic functions, Laplace transforms of Heaviside unit step function,Laplace transforms of Dirac delta functionWeek 8: Applications of Laplace transforms-I, Applications of Laplace transforms-II, Applications of Laplace transforms-III, Z – transform and inverse Z-transform of elementary functions, Properties of Z-transforms-IWeek 9: Properties of Z-transforms-II, Initial and final value theorem for Z-transforms, Convolution theorem for Z- transforms, Applications of Z- transforms--I,Applications of Z- transforms-IIWeek 10: Applications of Z- transforms--III, Fourier series and its convergence--I, Fourier series and its convergence--II, Fourier series of even and odd functions,Fourier half-range seriesWeek 11: Parsevel’s Identity, Complex form of Fourier series, Fourier integrals, Fourier sine and cosine integrals, Fourier transformsWeek 12: Fourier sine and cosine transforms,Convolution theorem for Fourier transforms,Applications of Fourier transforms to BVP-I,Applications of Fourier transforms to BVP-II,Applications of Fourier transforms to BVP-III